Let \(i_{\text{max}}(G)\)  denote the number of maximal independent sets of a graph \(G\) , i.e., the number of independent sets that are not contained in any larger independent set. Answering a question on the number of (maximal) cliques posed by Erdős and Moser, Moon and Moser [1]}, independently from Miller and Muller [2]} (see also [3]} for a short alternative proof) proved (by considering the complement) that for a graph \(G\)  of order \(n\) , where \(n \ge 2\) ,
\(i_{\text{max}}(G) \le {\left\lbrace \begin{array}{ll}3^{\frac{n}{3}} & \text{ if } n\equiv 0 \pmod {3},\\4\cdot 3^{\frac{n-4}{3}} & \text{ if } n\equiv 1 \pmod {3},\\2\cdot 3^{\frac{n-2}{3}} & \text{ if } n\equiv 2 \pmod {3}.\\\end{array}\right.}\) 
